﻿[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:00.27,0:00:04.04,Default,,0000,0000,0000,,These are the correct factors for each part of the fractions. I know factoring Dialogue: 0,0:00:04.04,0:00:07.59,Default,,0000,0000,0000,,can be tough, so if you're getting at least half of these right, great work. If Dialogue: 0,0:00:07.59,0:00:11.31,Default,,0000,0000,0000,,you got stuck on one of these, try to look back at your work and see where you Dialogue: 0,0:00:11.31,0:00:15.31,Default,,0000,0000,0000,,went wrong to get these factors. If you can't find an error in your work then Dialogue: 0,0:00:15.31,0:00:19.50,Default,,0000,0000,0000,,stay with me for this solution. For our first numerator we can pull out a 2x Dialogue: 0,0:00:19.50,0:00:25.19,Default,,0000,0000,0000,,from each term. So we'll have 2x times x squared minus 6x plus 9. Then we notice Dialogue: 0,0:00:25.19,0:00:29.63,Default,,0000,0000,0000,,that this is a special factoring pattern. It's a perfect square trinomial. So we Dialogue: 0,0:00:29.63,0:00:35.56,Default,,0000,0000,0000,,have 2x times x minus 3, times x minus 3. This gives us our first numerator. For Dialogue: 0,0:00:35.56,0:00:40.12,Default,,0000,0000,0000,,this denominator, we want to find the factors of negative 12, that sum to Dialogue: 0,0:00:40.12,0:00:44.06,Default,,0000,0000,0000,,negative 4. This allows us to rewrite our middle term. And then we'll use Dialogue: 0,0:00:44.06,0:00:48.58,Default,,0000,0000,0000,,factoring by grouping to get 3x plus 2 times x minus 2. This is our factored Dialogue: 0,0:00:48.58,0:00:53.36,Default,,0000,0000,0000,,form for our first denominator. For this numerator we want to find the factors Dialogue: 0,0:00:53.36,0:00:58.44,Default,,0000,0000,0000,,of 12 that sum to negative 8. These two factors are negative 6, 6 and negative Dialogue: 0,0:00:58.44,0:01:03.93,Default,,0000,0000,0000,,2. We'll use factoring by grouping to get 3x minus 2 time x minus 2 for this Dialogue: 0,0:01:03.93,0:01:09.40,Default,,0000,0000,0000,,numerator. And finally for this last denominator we pull out a 6x from both Dialogue: 0,0:01:09.40,0:01:14.36,Default,,0000,0000,0000,,terms leaving us with 6x times x squared minus 9. Then we can factor this Dialogue: 0,0:01:14.36,0:01:18.77,Default,,0000,0000,0000,,difference of squares using x plus 3 times x minus 3. This difference of squares Dialogue: 0,0:01:18.77,0:01:22.75,Default,,0000,0000,0000,,pattern appears in all sorts of math. So it's great that we can recognize it Dialogue: 0,0:01:22.75,0:01:23.36,Default,,0000,0000,0000,,quickly.